On quaternionic James numbers and almost-quaternion substructures on the sphere

In this paper a theorem about the relation between the divisibility of orders of obstructions to cross sectioning symplectic Stiefel manifolds and quaternionic James numbers is proved. As an application of this, the existence problem of almost-quaternion $k$-substructures on the sphere ${S^n}$ is solved for all $n$ and $k$ except for the case $n = 4m - 3$, $k = m - 1$ for some $m \geqslant 1$.